Well-posedness, global existence and large time behavior for Hardy–Hénon parabolic equations. (March 2017) - Ben Slimene, Byrame; Tayachi, Slim; Weissler, Fred B.
Abstract: In this paper we study the nonlinear parabolic equation ∂ t u = Δ u + a | x | − γ | u | α u, t > 0, x ∈ R N ∖ { 0 }, N ≥ 1, a ∈ R, α > 0, 0 < γ < min ( 2, N ) and with initial value u ( 0 ) = φ . We establish local well-posedness in L q ( R N ) and in C 0 ( R N ) . In particular, the value q = N α / ( 2 − γ ) plays a critical role. For α > ( 2 − γ ) / N, we show the existence of global self-similar solutions with initial values φ ( x ) = ω ( x ) | x | − ( 2 − γ ) / α, where ω ∈ L ∞ ( R N ) is homogeneous of degree 0 and ‖ ω ‖ ∞ is sufficiently small. We then prove that if φ ( x ) ∼ ω ( x ) | x | − ( 2 − γ ) / α for | x | large, then the solution is global and is asymptotic in the L ∞ -norm to a self-similar solution of the nonlinear equation. While if φ ( x ) ∼ ω ( x ) | x | − σ for | x | large with ( 2 − γ ) / α < σ < N, then the solution is global but is asymptotic in the L ∞ -norm to e t Δ ( ω ( x ) | x | − σ ) . The equation with more general potential, ∂ t u = Δ u + V ( x ) | u | α u, V ( x ) | x | γ ∈ L ∞ ( R N ), is also studied. In particular, for initial data φ ( x ) ∼ ω ( x ) | x | − ( 2 − γ ) / α, | x | large, we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity.